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GENERALIZED FIBONACCI NUMBERS OF THE FORM 11x2 + 1

  • Ogut, Ummugulsum ;
  • Keskin, Refik (Department of Mathematics, Sakarya University)
  • Received : 2017.11.27
  • Accepted : 2018.02.13
  • Published : 2018.03.25

Abstract

Let $P{\geq}3$ be an integer and let ($U_n$) denote generalized Fibonacci sequence defined by $U_0=0$, $U_1=1$ and $U_{n+1}=PU_n-U_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equation $U_n=11x^2+1$. We show that only $U_1$ and $U_2$ may be of the form $11x^2+1$.

Keywords

Generalized Fibonacci numbers;Generalized Lucas numbers;Congruences;Diophantine equation

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